Stochastic Volatility and Seasonality in ... - Interconti, Limited

Stochastic Volatility and Seasonality in Commodity
Futures and Options: The Case of Soybeans
Martin Christian Richter and Carsten Sørensen
Department of Finance
Copenhagen Business School
Solbjerg Plads 3
DK-2000 Frederiksberg
Denmark

Stochastic Volatility and Seasonality in Commodity
Futures and Options: The Case of Soybeans
Abstract
We estimate a continuous-time stochastic volatility model using panel data of soybean
futures and options on soybean futures. The model of commodity price dynamics
is within the class of so-called affine asset pricing models, and option prices
are determined using a standard inversion of characteristic functions approach. Our
modeling acknowledges that commodities exhibit seasonality patterns in both spot
price level and volatility and, hence, the involved commodity dynamics involve
stochastic differential equations that are inhomogeneous in time. The estimation
method is based on a state space formulation of the model and a quasi maximum
likelihood approach. Estimation results are obtained based on weekly observations
of soybean futures prices and options prices from the Chicago Board of Trade in the
period October 1984 to March 1999. Our empirical results support the conceptual
ideas in the theory of storage, though not the view that convenience yields are like
timing options.
JEL Classification: G13, C00

1 Introduction
This paper sets up and estimates a continuous-time model of commodity price behavior using
panel data of soybean futures prices and soybean option prices using data from the Chicago
Board of Trade (CBOT) over the period 1984-1999. The approach is conceptually related
to the analysis in Schwartz (1997), who sets up and estimates models of commodity futures
prices using the Kalman-filter and a state-space approach. 1 However, the model in this paper
includes additional realistic features since we, in particular, allow for stochastic volatility
and seasonality in the theoretical modeling of commodity price behavior. These features are
especially important for agricultural commodities with a harvesting pattern that induces a
seasonal component in the commodity price as well as in the volatility of the commodity price.
The seasonality feature in both commodity prices and volatilities is thus known to be one of the
empirical characteristics that make commodities strikingly different from stocks, bonds, and
other conventional financial assets; see e.g. the discussion in Routledge, Seppi and Spatt (2000).
Furthermore, as an innovative feature, our panel data estimation approach is based on both
futures prices as well as option prices across different exercise prices and maturities.
The commodity price behavior is modeled in continuous-time as part of a system of stochastic
differential equations within the class of asset pricing models where the drift and diffusion
coefficients are affine functions of the relevant basic state-variables. In our framework there are
three basic state-variables: the commodity spot price, the convenience yield, and the volatility
of the spot price. Affine models are tractable for asset pricing purposes and, in our case,
for the valuation of commodity futures contracts and options written on commodity futures.
Due to the tractability of affine models, they are widely used for modeling of term structures
of interest rates and pricing options on equity and foreign exchange rates. 2 In particular, it
is in general possible to determine relevant conditional characteristic functions within affine
models which in turn facilitates the evaluation of option prices through an inversion approach,
building on the ideas suggested by Heston (1993) for a specific stochastic volatility model of
1 See also Schwartz and Smith (2000) and Sørensen (2001) for empirical analyzes of commodity price models
following this approach.
2 See e.g. Duffie and Kan (1996) for a general specification of affine term structure models, Heston (1993)
and Bakshi, Cao and Chen (1997) for affine models of equity returns, and Backus, Foresi and Telmer (1996) and
Bates (1996) for affine models of foreign exchange rates. Duffie, Pan and Singleton (2000) provide a discussion
of other applications as well as a general treatment of the affine class of models, with the possibility of jumps
included.
1

currency and equity returns. The modeling of commodity spot price behavior in this paper
differs from models of interest rates, equity returns, and foreign exchange by the fact that
the involved stochastic differential equations are inhomogeneous in time since the drift and
diffusion coefficients are functions of calender time, due to the seasonality feature, and by
the inclusion of a convenience yield. On the other hand, we build on the same traditional
no-arbitrage approach to the pricing of commodity derivatives as used in analyzing the term
structure of interest, equity derivatives, and foreign exchange derivative, and also used widely
for analyzing commodity contracts and commodity hedging in related papers. 3
The estimation approach is based on quasi maximum likelihood and the panel data estimation
approach suggested originally by Chen and Scott (1993) in the context of models of
the term structure of interest rates. In particular, at each sample date we observe a panel
of futures prices and option prices which are related to the basic state-variables through a
non-linear measurement equation. The measurement equation is based on the pricing results
that we establish for the specific affine model of commodity price behavior with respect to
relevant futures and option prices. Following the ideas in Chen and Scott (1993), we assume
that all but three contracts are observed with measurement error. The latent value of the
basic state-variables can be exactly filtered out at each sample date by inversion based on the
three contract prices which are observed without error, and the likelihood of the full sample
can then be described based on the likelihood of the filtered paths of the state-variables. In the
evaluation of the likelihood of the filtered state-variables we apply a quasi maximum likelihood
approach where the relevant densities are substituted by Gaussian densities with appropriate
first- and second order moments. The estimation approach is analogues to the estimation
approach used in related papers on interest rates and equity returns in affine asset pricing
models. 4
3 See e.g. Black (1976), Brennan and Schwartz (1985), McDonald and Siegel (1986), Gibson and
Schwartz (1990), Jamshidian and Fein (1990), Bjerksund (1991), Cortazar and Schwartz (1994), Dixit and
Pindyck (1994), Schwartz (1997), Hilliard and Reis (1998), Miltersen and Schwartz (1998), and Schwartz and
Smith (2000) for advances on no-arbitrage models with a focus on the valuation of commodity contingent claims,
such as real options, and hedging.
4 See e.g. Chen and Scott (1993) and Dai and Singleton (2000) for estimation of models of the term structure of
interest rates, Duffie and Singleton (1997) for estimation of a model of swap yields, Duffie and Singleton (1999)
and Duffee (1999) for models of credit risky bonds, and Bates (2000) and Chernov and Ghyssel (2000) for
estimations of equity models using options data. This list of related papers concerned with estimation of affine
asset pricing models is very partial, and this is a fast growing research area. Additional relevant references on
2

The estimated commodity price model describes the simultaneous dynamics of the commodity
spot price, the convenience yield, and the volatility of the commodity spot price. When
discussing our specific estimation results, based on soybean futures and option contracts from
CBOT, we are in particular interested in how the results relate to the basic ideas of the theory
of storage by Kaldor (1939), Working (1948, 1949), Brennan (1958), and Telser (1958). The
theory of storage explains the differences between the spot and futures prices with different
contract horizons in terms of advantages of physical inventory compared to advantages of
ownership of a futures contract for future delivery of the commodity. Physical inventory is
associated with cost of carry such as storage costs and the interest forgone by investing in
storage. On the other hand, physical inventory gives rise to a convenience yield from being
able to profit from temporary price increases due to temporary shortages of the particular
commodity or from being able to maintain a production process despite abrupt shortages of
the commodity used as input. 5 The theory of storage predicts a negative relationship between
supply/inventories and convenience yields. Thus, when inventories are full and supply is plenty
the convenience yield from marginal storage is low, and vice versa when inventories are close
to being empty. 6 Furthermore, since holding inventory allows for flexibility in the timing of
the use of a given commodity, the convenience yield may be viewed as a timing option. Recently,
Routledge, Seppi and Spatt (2000) has thus provided a formal model supporting this
idea building on the apparatus of Wright and Williams (1989), Williams and Wright (1991),
Deaton and Laroque (1992, 1996), Chambers and Bailey (1996), among others, who formalize
the theory of storage in competitive rational expectations models.
Our empirical results are to some degree in accordance with the theory of storage. For
example, the seasonal components in both the soybean convenience yields and volatilities are
at the highest just before harvesting when inventories are low and supply scarce. The spot
price is estimated to be positively correlated with the convenience yield, so this is also a period
where spot prices are relatively high. While this supports the ideas in the theory of storage,
related literature can be found in e.g. Singleton (2000) and Pan (2001).
5 When we refer to the convenience yield in the following and in the formal continuous-time model, we are
in fact referring to the net convenience yield, i.e. the convenience yield net of storage costs, unless otherwise
specified.
6 Since inventories/supply of agricultural commodities vary with the season, a seasonality pattern in convenience
yields is, therefore, well in accordance with the prediction of the theory of storage, as discussed by e.g.
Fama and French (1987).
3

our results on the other hand do not support the view that the convenience yield behaves
like a timing option. In particular, our estimation results suggest that convenience yields and
volatilities are slightly negatively correlated contrary to how an option depends on volatility.
Furthermore, these estimation results are consistent across different time periods.
Furthermore, our empirical results demonstrate that the soybean commodity spot price is
positively correlated with volatility. This is opposed to the so-called leverage effect in equity
markets where the correlation between stocks and volatility is usually estimated negative. This
estimation result is, however, consistent with the observation that the implied volatility “smile”
or “smirk” in soybean options has opposite slope of the post-1987 crash “smirk” observed for
equity options.
The paper is organized as follows. In section 2, the model and estimation approach is
described in details and in section 3 the data and estimation results are presented. A final
section concludes.
2 The model and estimation approach
Let W = (W 1 , W 2 , W 3 ) be a three-dimensional Brownian motion defined on a probability space
(Ω, F, P). F = {F t : t ≥ 0} denotes the standard filtration of W and, formally, (Ω, F, F, P)
is the basic model for uncertainty and information arrival in the following. We will assume
that W has a constant “instantaneous” correlation matrix Σ with entries ρ ij and ones in the
diagonal.
2.1 Commodity price dynamics
The model of commodity price behavior incorporates important realistic features such as
stochastic convenience yields and stochastic volatility as well as a seasonal feature. Let P t
denote the spot commodity price at time t. The convenience yield and the seasonal adjusted
spot price volatility at time t are denoted δ t and v t , respectively. The dynamics of the threedimensional
process (P, δ, v) are described by the following stochastic differential equation
system,
dP t = P t
[
(r − δ t + λ P e ν(t) v t ) dt + e ν(t)√ v t dW 1,t
]
dδ t = (α(t) − βδ t + λ δ σ δ e ν(t) v t ) dt + σ δ e ν(t)√ v t dW 2,t (2)
dv t = (θ − κv t + λ v σ v v t ) dt + σ v
√
vt dW 3,t (3)
(1)
4

where β, κ, θ, λ P , λ δ , λ v , σ δ , and σ v , as well as the correlation parameters of W , are constant
parameters that we will estimate in the subsequent sections. α(t) and ν(t) are deterministic
functions that aim at describing a seasonal pattern in convenience yields and volatilities; the
functional forms of α(t) and ν(t) are described and discussed below.
The parameters λ P , λ δ , and λ v are the risk premia on commodity price uncertainty, convenience
yield uncertainty, and volatility uncertainty. We will assume markets are complete and
the existence of an equivalent martingale measure, Q, so that prices on contingent claims can
be uniquely derived by evaluating the relevant expectations of the discounted payoff under Q.
Using a standard terminology, we will also refer to Q as the risk-neutral probability measure
in the following. Under the risk-neutral probability measure, risk premia are zero and the
dynamics of (P, δ, v) are described by,
dP t = P t
[(r − δ t ) dt + e ν(t)√ v t dW Q 1,t
dδ t = (α(t) − βδ t ) dt + e ν(t) √
σ δ vt dW Q 2,t (5)
√
dv t = (θ − κv t ) dt + σ v vt dW Q 3,t (6)
From equations (1) and (4) it follows that the convenience yield, δ t , can be interpreted as
the return shortfall on an inventory position of the specific commodity. 7 Basically, in order
to break-even on the marginal inventory position, there must be a gain, or convenience yield,
from holding inventory to exactly offset the return shortfall.
The parameters β and κ determine the degree of reversion to the deterministic seasonal
pattern in the convenience yield and the long-run volatility level θ, respectively. Likewise,
σ δ and σ v are parameters that, together with the level of v t , determine the volatility of the
convenience yield and the volatility of the stochastic volatility of the commodity price. The
parameter r is the short default-free interest rate which is assumed constant (but we allow to
vary over time in our empirical analysis). 8
The deterministic seasonal functions α(t) and ν(t) are defined as,
∑
α(t) = α 0 + (α k cos (2πkt) + αk ∗ sin (2πkt)) (7)
K α
k=1
7 In fact, the convenience yield may be defined as the return shortfall on holding the commodity; see, e.g.,
MacDonald and Siegel (1984).
8 Our pricing results will carry through if interest rates are stochastic but distributed independently of the
soybean dynamics. In this case, discounting is done using the appropriate zero coupon bonds, as in our empirical
approach.
]
(4)
5

and
∑K ν
ν(t) = (ν k cos (2πkt) + νk ∗ sin (2πkt)) (8)
k=1
where K α and K ν determine the number of terms in the sums and α 0 , α k , αk ∗, k = 1, . . . , Kα ,
and ν k , νk ∗, k = 1, . . . , Kν are constant parameters to be estimated. The specific functional
form of the deterministic seasonal components were originally suggested by Hannan, Terrell,
and Tuckwell (1970) as an alternative to standard dummy variable methods in econometric
modeling of seasonality. Due to the continuous time feature of our analysis this is a flexible
and natural choice of modeling the seasonal aspects of commodity price behavior which is also
applied in Sørensen (2001).
2.2 Futures prices
Following Cox, Ingersoll and Ross (1981), the futures price at time t on a futures contract
for delivery at time T ≥ t can be obtained by taking the relevant expectations of the future
spot price, F t (T ) = E Q t [P T ]. Equivalently, by using the Feynman-Kac formula, F t (T ) =
F (P t , δ t , v t , t; T ) where F (·) can be found as the solution to a partial differential equation on
the form
1
2 e2ν(t) vP 2 ∂2 F
+ 1 ∂P 2 2 σ2 δ e2ν(t) v ∂2 F
∂δ 2
+ ρ 23 σ δ σ v e ν(t) v ∂2 F
∂F
∂δ∂v
+ (r − δ)P
∂P
+ 1 2 σ2 vv ∂2 F
∂v 2
+ ρ 12 σ δ e 2ν(t) vP ∂2 F
∂P ∂δ + ρ 13σ v e ν(t) vP ∂2 F
∂P ∂v
+ (α(t) − βδ)
∂F
∂δ
+ (θ − κv)
∂F
∂v + ∂F
∂t
= 0
(9)
with terminal condition F (P, δ, v, T ; T ) = P .
For a given and fixed expiration date T , the solution to the partial differential equation in
(9) is given by
F (P, δ, v, t; T ) = P e A(t;T )+B(t;T )v+D(t;T )δ (10)
where
D(t; T ) = − 1 β
(
1 − e −β(T −t)) (11)
and A(t; T ) and B(t; T ) are the solutions to the ordinary differential equations
1
2 σ2 δ e2ν(t) (D(t; T )) 2 + ρ 12 σ δ e 2ν(t) D(t; T ) + 1 2 σ2 v (B(t; T )) 2
+ (ρ 13 σ v e ν(t) + ρ 23 σ δ σ v e ν(t) D(t; T ) − κ)B(t; T ) + B ′ (t; T ) = 0 , B(T ; T ) = 0
(12)
and
r + α(t)D(t; T ) + θB(t; T ) + A ′ (t; T ) = 0 , A(T ; T ) = 0 (13)
6

Solutions to the ordinary differential equations in (11) and (13) are not available in closed form.
Solutions are, however, easily obtained numerically with very high precision and speed using
for example Runge-Kutta methods; this is the approach followed in our empirical analysis.
2.3 Option prices
We will consider options written on commodity futures. By using Ito’s lemma on the futures
price, as described in equation (10), it can be seen that the dynamics of the futures price under
the equivalent martingale measure, Q, are given by
dF t (T ) = F t
√
vt
[
e ν(t) dW Q 1,t + σ vB(t; T ) dW Q 3,t + σ δe ν(t) D(t; T ) dW Q 2,t . ]
Let C t be the price of a European call option written of a the futures price of a futures contract
expiring at time T . The option has strike price K and matures at time τ; t ≤ τ ≤ T . The call
price can be determined by taking the relevant expectations,
C t = E Q t
(14)
[
]
e −r(τ−t) max[0, F τ (T ) − K] . (15)
From the Feynman-Kac’s formula, and the dynamics of F t and v t in equations (14) and (6), it
follows that the option price can be represented as C t = C(F t , v t , t) where the function C(·)
is the solution to the partial differential equation
where
1
2 σ2 F (t; T )vF 2 ∂2 C
+ 1 ∂F 2 2 σ2 vv ∂2 C
+ σ
∂v 2 F v (t; T )vF ∂2 C
∂C
∂F ∂v
+ (θ − κv)
∂v + ∂C
∂t − rC = 0 (16)
σ 2 F (t; T ) = e2ν(t) + σ 2 v (B(t; T )) 2 + σ 2 δ e2ν(t) (D(t; T )) 2 + 2ρ 13 σ v e ν(t) B(t; T )
+ 2ρ 12 σ δ e 2ν(t) D(t; T ) + 2ρ 23 σ v σ δ e ν(t) B(t; T )D(t; T )
and
σ F v (t; T ) = ρ 13 σ v e ν(t) + σ 2 vB(t; T ) + ρ 23 σ v σ δ e ν(t) D(t; T )
and with terminal condition C(F, v, τ) = max[0, F − K]. σ F (t; T ) describes the volatility on
the underlying futures price whereas σ F v (t; T ) describes the “instantaneous” rate of covariance
between the futures price and the stochastic volatility process.
This is basically the Heston (1993) model but with σF 2 (t; T ) and σ F v(t; T ) being functions
of time instead of constants (and the underlying being a futures price instead of a spot price).
7

Hence, using the arguments and approach in Heston (1993), pp. 330-331 and his Appendix,
one obtains the following option pricing formula on the form of Black (1976),
C(F, v, t) = e −r(τ−t) [F P 1 − KP 2 ] (17)
where (see equation (18) in Heston (1993))
P j = 1 2 + 1 ∫ [
]
∞
e −iφ ln[K] f j (x, v, t; φ)
Re
dφ , j = 1, 2 (18)
π
iφ
0
and where x = ln F and f j (·; φ), j = 1, 2, denote characteristic functions that are obtained
as solutions to PDE’s with terminal condition e iφx . Since coefficients in the particular PDE’s
are affine and the terminal conditions are exponential-affine, the solutions for f j , j = 1, 2, are
exponential-affine and on the form
f j (x, v, t; φ) = e a j(t)+b j (t)v+iφx , j = 1, 2 (19)
where a j (t) and b j (t) are solutions to the ordinary differential equations
b ′ j (t) = ( 1
2 φ2 − u j iφ ) σ 2 F (t; T ) + (k j(t) − iφσ F v (t; T )) b j (t) − 1 2 σ2 v (b j (t)) 2 , b j (τ) = 0 (20)
and
a ′ j(t) = −θb j (t) , a j (τ) = 0 (21)
with u 1 = 1 2 , u 2 = − 1 2 , k 1(t) = κ − σ F v (t; T ), and k 2 (t) = κ.
2.4 The estimation approach
The estimation approach is based on formulating the model in state space form so that the
model is represented by a measurement equation, and a transition equation. In our case
the state space model is non-linear and the specific approach relies on a general state space
estimation approach suggested originally by Chen and Scott (1993) for estimation of models
of the term structure of interest rates.
The data are in the following observed at equidistant time points, t n , n = 0, . . . , N, and
∆ = t n+1 − t n . The transition equation describes the dynamics of the basic unobserved
state-variables. In our case, there are three state-variables which can be summarized by the
three-dimensional state-vector X n = (p tn , δ tn , v tn ) where p t = log P t . The transition equation
is given by the discrete-time solution to the stochastic differential system in (1), (2), and (3)
between the observation time points.
8

The measurement equation relates the state-variables to the observed (log-) futures prices
and option prices. Following the approach in Chen and Scott (1993), we will assume that three
prices on soybean contingent claims are observed without measurement error. In particular,
this allows one to filter out the three unobserved state-variables by solving three equations
with three unknowns at all observation time points. Let the vector Y n denote the panel-data
observation of (log-) futures prices and options prices at time t n , n = 0, . . . , N.
We have
Y ′ n = (Y ′
(1),n , Y ′
(2),n ) where Y (1),n is the three-dimensional vector of contract prices observed
without measurement error while Y (2),n is the m-dimensional vector of contract prices that are
observed with measurement error. Formally, the measurement equation is given by
⎛ ⎞ ⎛ ⎞ ⎛ ⎞
⎝ Y (1),n
⎠ = ⎝ f (1),n(X n )
⎠ + ⎝ 0 ⎠ (22)
Y (2),n f (2),n (X n ) ɛ n
where 0 is a three-dimensional vector of zeros and ɛ n , n = 0, 1, . . . , N, are serially uncorrelated
measurement error terms that are assumed identical m-dimensional normally distributed with
mean zero and variance-covariance matrix Γ; f (1),n and f (2),n are a three-dimensional valued
function and an m-dimensional valued function, respectively, that return the relevant theoretical
value of the logarithm to the futures prices and options prices, as implied by the futures
and options pricing expressions in (10) and (17).
In our implementation, we assume that a short maturity at-the-money call option, the
corresponding futures price, and the futures price on a long maturity futures contract are
observed without measurement error; i.e. these are the contract prices that enter Y (1),n at the
different sample dates. (Y (2),n has dimension m = 10 in our implementation since six (log-)
futures prices and four option prices are assumed observed with measurement noise.) Having
observed a total of three futures and option prices without measurement noise, it is possible to
filter out the three-dimensional state-vector X n by simply inverting the relationship between
Y (1),n and X n , as given by the pricing expressions in (10) and (17) that are the entries in f (1),n .
Specifically, we have the relationship
X n = f −1
(1),n (Y (1),n). (23)
Whenever we refer to the value of X n in the following, it is assumed that X n is observed
through the filtering equation (23).
Let ψ denote the set of parameters entering the description of state-variable dynamics in
(1), (2), and (3) and in the pricing results in (10) and (17). We are interested in estimating the
9

parameters in ψ as well as the variance-covariance matrix Γ of the measurement error terms in
the measurement equation (22). Following Chen and Scott (1993), the log-likelihood function
has the following form:
l(Y 0 , Y 1 , . . . , Y N ; ψ, Γ) = ∑ [
N
∣
n=1
log p(X n |X n−1 ) − log
∣
∣ ∂f (1),n
∂X
− m 2 log(2π) − 1 2 log |Γ| − 1 2 ɛ′ nΓ −1 ɛ n
]
.
The first term on the right-hand side of (24) is the log-likelihood value of the implied statevariables
as filtered through the relationship in (23), and p(X n |X n−1 ), n = 1, . . . , N, denote
the relevant conditional densities of the Markovian discrete-time solution to the stochastic
differential equation system (1), (2), and (3) which describes the transition equation. 9
second term on the right-hand side of (24) is the Jacobian determinant of the transformation
between the observed contract prices in Y (1),n and X n . The last terms on the right-hand side
of (24) are the likelihood of the normally distributed measurement errors.
The estimation results obtained in the following analysis is obtained by numerical maximization
of the log-likelihood function in (24).
(24)
The
However, since the conditional densities
p(X n |X n−1 ), n = 1, . . . , N, are not known in closed-form analytical form, we have chosen
to rely on a quasi maximum likelihood approach where these conditional densities are substituted
by densities from the three-dimensional normal distribution; the appropriate first- and
second order moments follow from Lemma 1, as stated and proved in the appendix. Likewise,
in our calculation of standard errors on parameter estimates, we follow the general quasi maximum
likelihood approach suggested originally by White (1982); see also e.g. Hamilton (1994),
pp. 126,145.
The concrete implementation and optimization involves many numerical considerations
and, therefore, we have allocated the following subsection to this issue.
However, at this
point it can be noted that the variance-covariance matrix of the measurement errors, Γ, can
be concentrated out of the log-likelihood function.
From the first order conditions of the
maximization of (24), it is thus seen that the estimator of Γ (for a given set of parameters ψ)
has the form
ˆΓ = 1 N
N∑
ɛ n ɛ ′ n = 1 N
n=1
N∑ (
Y(2),n − f (2),n (X n ) ) ( Y (2),n − f (2),n (X n ) ) ′
n=1
(25)
9 Note that the involved stochastic differential equation system is inhomogeneous due to the seasonality
feature of the model, though, the time-dependence of the relevant conditional densities is suppressed in (24).
10

and, N ˆΓ ∼ W m (N, Γ) where W m (N, Γ) is the m-dimensional Wishart distribution with N degrees
of freedom. Substituting the expression for Γ in (25) into the log-likelihood function (24)
simplifies the numerical optimization problem since we only have to determine the maximum
likelihood parameter values of ψ with a numerical optimization routine. The numerical optimization
is done in GAUSS using the method of Broyden, Fletcher, Goldfarb and Shanno
(BFGS). We use the standard GAUSS options in the numerical BFGS optimization algorithm.
2.5 Numerical implementation
In our implementation of the above estimation approach we observe data at a weekly frequency
and at a total of 753 sample dates. At each sample date we have a panel-observation consisting
of eight (log-) futures prices and five option prices. The evaluation of the log-likelihood
function, hence, involves the evaluation of a large number of theoretical futures and options
prices using the formulas stated in (10) and (17). In this section we will briefly describe how
these pricing expressions are implemented and how the log-likelihood function is calculated in
a relatively fast but still accurate way. Using the implementation described in the following,
the evaluation of a single value of the log-likelihood function in GAUSS takes about one minute
on a Pentium III 500 MHz computer while the evaluation of a single option price on average
is done in about one hundredth of a second. A single iteration of the BFGS algorithm takes
about one hour, and the parameter estimates presented in a subsequent section is obtained
using a total of about three weeks of computer time.
The evaluation of a futures price as in (10) requires solving the system of ordinary differential
equations in (12) and (13). The numerical solution to (12) and (13) is implemented using
the classical Runge-Kutta method of fourth order accuracy combined with Richardson extrapolation.
The discrete time steps used in the classical Runge-Kutta method are always chosen
so as not to exceed five days or, equivalently, (5/365.25=) 0.01369 years. In the Richardson
extrapolation this solution is combined with the Runge-Kutta solution with half as many time
steps by using the relevant Richardson extrapolation formula for numerical methods of fourth
order accuracy.
The evaluation of an option price as in (17) requires the numerical evaluation of integrals of
the form in (18). The numerical integration is implemented by the Gauss-Laguerre quadrature
formula using twenty points in the evaluation of the integrand function. Each evaluation of
such an integral thus requires solving numerically the system of ordinary differential equations
11

in (20) and (21) for twenty different values of φ as well as simultaneously solving the ordinary
differential equation for B(·) in (12) since the coefficients σ 2 F (·) and σ F v(·) in (20) are
functions of B(·). This system of forty one (= 20 + 20 + 1) first order differential equations
is solved simultaneously using the classical Runge-Kutta method combined with Richardson
extrapolation following exactly the same procedure as described for futures prices above.
Evaluation of the log-likelihood function in (24) requires inverting the relationship between
the unobserved state-vector X n and the observed prices in Y (1),n , as described conceptually by
the filtering equation (23). In our implementation, we assume that a short maturity at-themoney
call option, the corresponding futures price, and the futures price on a long maturity
futures contract are observed without measurement error.
The inversion is carried out by
first solving for the implied value of the volatility, v tn , in the call option price by solving one
equation with one unknown by the Newton-Raphson method (in this step we use that the
underlying futures price is observed without noise). In solving for the implied volatility by the
Newton-Raphson method, we use that
[
∂C
∂v = e−r(τ−t) F ∂P 1
∂v − K ∂P ]
2
∂v
and
∂P j
∂v = 1 π
∫ ∞
0
Re
[
b j (t) e−iφ ln[K] f j (x, v, t; φ)
iφ
]
(26)
dφ , j = 1, 2 (27)
which follows by differentiating the call option expression in (17) and (18) for a fixed value
of the underlying futures price. Evaluating the relevant derivatives in this step thus require
evaluation of integrals of the same form as in the evaluation of options prices, and the same
procedure is applied in this case. 10 Having solved for the implied value of the volatility statevariable,
we use the exponential affine structure of the futures prices (i.e. the log-futures prices
are affine functions of the state-variables) to solve for the values of the two remaining statevariables
p tn and δ tn by simply solving two linear equations with two unknowns. Furthermore,
the Jacobian determinant of the transformation between Y (1),n and X n is calculated as a byproduct
of the numerical calculations in this inversion approach since we have that
∂f (1),n
∣ ∂X ∣ = ∂C
∂v (D(t n; T s ) − D(t n ; T l )) (28)
10 It can be noted that the relevant system of forty one first order differential equations must only be solved
once at each sample date in order to determine the relevant derivatives in the different Newton-Raphson steps
as well as the calculation of all option prices with the same maturity.
12

where T s and T l refer to the relevant maturities of the involved short and long futures contract
and D(·) is defined in (11).
Finally, the relevant conditional means and conditional variances that enter the normal
density functions in the quasi maximum likelihood approach are determined by solving numerically
the integrals in (30) and (31), as stated in Lemma 1 in the appendix. Since data are
observed at a weekly frequency we have ∆ = 1/52 = 0.01923 years. The numerical integration
is in this case done by using the trapezoidal rule and using the second-order Runge-Kutta
method (Heun’s method) with one time step to solve the differential equation in (32) and
obtain the value of Φ in the right-end-point of the integrals. Many of the entries in (30) and
(31) can be calculated in explicit analytical form, but not all, and the suggested approximation
is very accurate compared to for example a left-end-point approximation of the integrals. The
later approximation corresponds to the case where the relevant normal densities in the quasi
maximum likelihood approach are obtained by an Euler approximation of the continuous-time
differential equation system in (1), (2), and (3). In our estimations we originally obtained
starting parameter values using such an Euler approximation, but with the specific data the
estimates and the likelihood of observations are not ignorably altered by going to a more
accurate approximation of the relevant moments in the conditional densities. 11
3 Empirical results
In this section we will describe the soybean futures and options data and, subsequently, present
the formal estimation results based on the model and estimation approach presented above.
3.1 Data
The data consists of weekly observations of soybean futures prices and options on soybean
futures from CBOT in the period from the beginning of soybean options trading on CBOT in
October 1984 and to March 1999. The futures prices and option prices involved are settlement
prices at CBOT for Wednesdays (or Tuesday if Wednesday is unavailable). In fact, settlement
prices are available on a daily basis but the weekly sampling frequency is chosen mainly to
reduce problems due to microstructure issues such as daily price limits imposed by CBOT.
11 In particular, the likelihood is significantly affected for values of the volatility state-variable being very close
to zero.
13

The Futures data
The CBOT soybean futures have seven expiration months in the sampling period: January,
March, May, July, August, September, and November. At any particular date, more than
seven soybean futures contracts may be traded since for example futures with expiration in
July this year and next year may be traded simultaneously. The maximum number of futures
contracts that are open at all the dates in our sample is eight. In order to have same number
of futures observations at each sample date, we have thus chosen to only include the eight
futures contracts with the shortest time to expiration at any sample date.
In the estimation approach we need the times to maturity for the different futures contracts
at the different sample dates. The contractual terms at CBOT specify that physical delivery
(in the form of an inventory receipt) may occur at any business day throughout the expiration
month and, hence, that the last delivery day is the last business day of the month. However,
the last trading day of the futures contract is the seventh business day preceding the last
business day of the delivery month. Since the last trading day is the last day that a contract
can in principle be closed at CBOT without physical delivery, we have chosen this day as the
expiration date when constructing the times to maturity for the involved futures contracts in
the estimations presented in the next section.
Table 1 provides summary statistics with respect to the involved soybean futures contracts.
[ INSERT TABLE 1 ABOUT HERE ]
The table states the average number of days to maturity and mean and standard deviation
of all the soybean futures prices in the dataset. In addition, the tables provide summary
statistics for the futures contracts categorized into expirations months as well as the time to
maturity of the contracts. In the following, the terminology “1. Maturity” is used as notation
for the futures contract that has the shortest time to maturity at a given sample date; the “2.
Maturity” represents the futures contract with the second shortest time to maturity, and so
on.
The table suggests at least three features of the soybean futures prices that are captured by
the model in section 2. (i) Futures prices display a seasonal pattern since the soybean futures
price is on average high in July prior to the US harvesting and low in November after the US
harvesting. (ii) The variability of futures prices also seem to have a seasonal pattern since the
standard deviations for the futures prices, when grouped into expiration months, display the
same time pattern as for the level of futures prices. And, (iii) the variability of long futures
14

prices seems lower than that of short futures prices. The lower variation of futures prices with
a long time to maturity compared to futures prices with a short time to maturity is suggested
by the different standard deviations for the futures prices grouped into their time to maturity
in Table 1. The tabulated standard deviations are thus monotonically decreasing for increasing
maturities. This pattern is consistent with the so-called “Samuelson hypothesis” that predicts
that long futures prices are less volatile than futures prices on contracts with short time to
maturity.
The time series aspects of the soybean futures data are illustrated in Figure 1.
[ INSERT FIGURE 1 ABOUT HERE ]
In the figure we have graphed the time-series for the futures contracts with the shortest time to
maturity and the futures contracts with the longest time to maturity over the sample period. A
visual inspection of the figure suggests that the time series of the futures price on the shortest
maturity contract is more volatile than the futures price on the longest maturity contract. In
particular, if one enter a horizontal line at the overall futures price mean at 627.49 cents per
bushel (cf. Table 1) in the figure, the deviations from this overall mean is more significant
for the short maturity futures price over the sample period. Again, this is in line with the
“Samuelson hypothesis” and a feature of the soybean futures prices which is captured in the
formal modeling in section 2 by including a mean-reverting convenience yield.
The options data
The involved CBOT soybean option contracts are options written on soybean futures prices.
The relevant futures contracts expire in: January, March, May, July, August, September,
and November. The option contract expires around one month prior to the expiration of the
underlying futures contract. Specifically, according to the CBOT contract specification, the
last trading day is the Friday which precedes by at least five business days the last business
day of the month preceding the option month. The expiration date is the Saturday following
the last trading day.
Both call options and put options are traded. Also, at any particular sample date a variety
of options contracts, which differ with respect to the underlying futures and with respect to
the strike prices, are quoted. However, in our sample we have only included five call option
contracts at each sample date. The specific call option contracts are selected in order to
represent very liquid contracts. The sample thus includes three call option contracts that
15

have the shortest possible time to maturity (though, we ignore very short contracts so that
all the considered options have at least 15 days to maturity). The three call options with a
short maturity are an at-the-money (ATM) option, an out-of-the-money (OTM) option, and
an in-the-money (ITM) option. The ATM option is chosen as the call option that has a strike
price as close as possible to the current underlying futures price. The OTM call option is the
quoted call option with a strike price just higher than the ATM call option and, likewise, the
ITM call option is the quoted call option with a strike price just lower than the ATM call
option. Furthermore, our sample includes two longer term ATM options that are written on
the next maturing futures prices.
The CBOT options on soybean futures are American-style. Since our pricing formulas are
relevant only for European-style options, the option data are adjusted for the early exercise
premium. The early exercise premia is estimated using a standard Black-Scholes framework
and applying a trinomial lattice model combined with Richardson extrapolation, following
Broadie and Detemple (1996). 12 The relevant interest rates are obtained by interpolation of
3 months, 6 months, and 1 year to maturity T-bills available from the Federal Reserve H.15
release. The rates are quoted on a banker’s discount basis and converted appropriately; see
Jarrow (1996), chapter 2. The American-style CBOT option prices minus the estimated early
exercise premia are the European-style input data used in the empirical analysis.
Table 2 provides summary statistics on the involved input call option prices.
[ INSERT TABLE 2 ABOUT HERE ]
Instead of exhibiting summary statistics of the raw option prices, we have chosen to represent
the option data by implied volatilities in Table 2. The relevant implied volatilities of the
involved options on soybean futures are backed out using the Black (1976) formula. Three
features of the option data can be pointed out: (i) The implied volatilities have a clear seasonal
pattern, (ii) the implied volatilities for the ATM options are slightly increasing as a function
of time to maturity, and (iii) the volatilities on the short maturity contract suggest a “smile”
or “smirk” pattern. Note, however, that the “smirk” is downward sloping as a function of the
strike price in contrast to the upward sloping “smirk” in equity index options observed after
the October 1987 market crash; see, e.g., Rubinstein (1994).
In Figure 2 the time series properties of the implied volatilities for the shortest ATM option
are displayed.
12 The results in Broadie and Detemple (1996) suggest that this method is efficient with respect to accuracy.
16

[ INSERT FIGURE 2 ABOUT HERE ]
As suggested by Figure 2, the implied volatilities are quite variable over time. This, combined
with the clear seasonal pattern in implied volatilities displayed in Table 2, is our basic
motivation for the formal modelling of the stochastic volatility in section 2.
3.2 Estimation results
In this section we will present and discuss the estimation results based on the above data set
and the estimation approach described in section 2.4. We start by discussing the full-sample
estimates but, subsequently, we also present estimation results where the data is split into two
sub-samples in order to investigate the parameter stability of the model over time.
The parameter estimates obtained from the state space estimation approach and the full
sample of observations are tabulated in Table 3.
[ INSERT TABLE 3 ABOUT HERE ]
In the following discussion of the full-sample parameter estimates in Table 3, we will start
by focusing on the structural parameters of the soybean spot price volatility process and the
convenience yield process parameters and, subsequently, we discuss the implications of the
estimated measurement noise structure.
Stochastic volatility and seasonality
The volatility of the implied soybean spot price is determined by the process v t and the seasonal
function ν(t). In particular, as inferred from the spot price dynamics in (1), or the risk-neutral
analog in (4), e 2ν(t) v t is the rate of variance on the soybean spot price and, hence, e ν(t)√ v t is
the spot price volatility. Since the seasonal function ν(t) captures any regular and systematic
seasonal pattern in the spot price volatility, √ v t can be interpreted as the seasonally adjusted
spot price volatility. Under the risk-neutral measure, the long-run level for v t is estimated to be
0.0679 (= θ/κ) which implies a value of the long-run seasonal adjusted volatility of about 26.1%
(= √ 0.0679). Furthermore, the volatility process exhibits a strong degree of mean-reversion
since the parameter estimate of κ is significantly higher than zero. 13 The parameter estimate
13 Throughout the discussion we will refer to an estimate being significantly differently from a given value if
the given value is not within the estimate plus/minus two standard deviations (the usually applied asymptotic
95% confidence interval). The 95% confidence interval for κ is thus [1.7400, 2.8016].
17

of κ (= 2.2708) implies that the so-called half-life of shocks to the seasonal adjusted variance
rate v t is 0.305 years (= log 2/κ), under the risk-neutral probability measure. Basically, this
means that it takes about three to four months before a given shock in volatility (e.g. due to
changes in harvest expectations or temporary changes in demand for soybeans) is expected to
have levelled off to half of its immediate effect. Under the true probability measure, however,
the degree of mean-reversion is even stronger and estimated to be 5.0754 (= κ − λ v σ v ) which
corresponds to a half-life of shocks of 0.137 years, or about one and a half months. Moreover,
under the true probability measure the long-run level of the seasonal adjusted volatility, as
reflected in the estimates in Table 3, is 17.4% (= √ θ/5.0754). These long-run volatility levels
seem well in accordance with the overall average Black-Scholes implied volatility of 19.99%,
as tabulated in the summary statistics in Table 2.
The seasonal variation in volatilities is captured by the seasonal function ν(t) defined in (8).
This function is a sum of self-repeating trigonometric functional terms. In our estimation we
have on beforehand limited the number of terms to two in order to keep the model sufficiently
parsimonious for numerical optimization to be possible and, hence, in our implementation this
function is described by a total of four parameters: ν 1 , ν1 ∗, ν 2, and ν2 ∗.14
All the seasonal
parameters are estimated to be significantly different from zero. The seasonal pattern in
volatilities that the parameter estimates reflect is illustrated in Figure 3 where the seasonal
function ν(t) is plotted.
[ INSERT FIGURE 3 ABOUT HERE ]
The figure illustrates that the seasonal function has two local maxima and two local minima
(though, this feature is much more clear for the convenience yield seasonal function α(t), as
to be discussed below). The global maximum is achieved in late July about two months prior
to the beginning of the US harvest (which occur from mid to late September and through
October). 15 The flowering and pod filling of the plant, which is of crucial importance for
the final seed yield and soybean production occur around this time, and this is thus a period
14 We have estimated the model including additional terms in the seasonal functions ν(t) and α(t) (the analog
for the convenience yield process) but keeping all other parameters fixed at the estimated values in Table 3. This
did not affect the estimated seasonal patterns substantially. Also, it may be noted that in a similar analysis with
futures price observations and seasonality in convenience yields, Sørensen (2001) limits the relevant number of
terms involved in the seasonal function to two based on a formal criteria, the Akaike Information Criteria.
15 The global maximum of the seasonal function ν(t) is reached on July 21 where the function value is 0.365
while the global minimum is reached on March 23 where the function value is -0.249.
18

where adverse weather conditions may significantly affect next year’s US supply of soybeans.
At this time of the year the volatility in soybeans is estimated to be 44% (= e 0.365 − 1)
above “normal,” where normal refers to the case when ν(t) = 0. On the other hand, the
soybean volatility is low in March prior to the US planting (which occur in May and June)
and at its lowest the volatility is 22% (= 1 − e −0.249 ) below normal. As discussed above,
the seasonal adjusted volatility reverts over time to a long-run level of around 17.4%. The
estimated seasonal pattern, however, suggests that the mean-reverting level is closer to 25.1%
(= 1.44 × 17.4%) in July and 13.6% (= 0.78 × 17.4%) in March. These mean-reverting levels
are of the same magnitude, though slightly below, the average volatilities tabulated in Table 2.
Note in addition that in Table 2 the minimum average implied volatility is obtained for the
options written on the May futures while the maximum average implied volatility is obtained
for the option written on the September futures. But since these option contracts expire one
month prior in April and August, respectively, and since the average times to maturity are
78.49 days and 59.04 days, these option contracts are precisely the contracts that cover the
relevant periods through late March and late July, respectively.
Convenience yields and theory of storage
The parameters estimated for the convenience yield process in (2), (5), and (7) also reveal a
clear seasonal pattern. In particular, the convenience yield process reverts to a level that depends
on the season, as reflected in the seasonal function α(t). The degree of tendency towards
this “normal” seasonal level is described by the parameter β. Note that while the convenience
yield drift term has this simple interpretation under the risk neutral measure in (5), the dynamics
are more complicated under the true probability measure since the convenience yield
risk premium is assumed proportional to the time-varying volatility (including the seasonality
feature of volatility). However, since the convenience yield risk premium parameter λ δ is estimated
close to (and not significantly different from) zero, we will continue the discussion as if
λ δ = 0 or, equivalently, as if the risk-neutral dynamics and the true dynamics coincide. The
estimate of β is 0.8145 which imply a half-life of unexpected shocks to the convenience yield
process of about 0.851 years (= log 2/β). Hence, the mean-reverting tendency is seemingly
less strong than the similar feature in the volatility process, as discussed above. The seasonal
function α(t) is described by a total of five parameters: α 0 , α 1 , α1 ∗, α 2, and α2 ∗ . These are all
estimated to be significantly different from zero. The parameter α 0 is a constant and without
19

a seasonality feature the long-run level for the convenience yield process would thus be 0.0751
(= α 0 /β); we interpret this as the “normal” level of the convenience yield when adjusted for
seasonality.
The seasonal pattern of the mean-reverting level of the convenience yield is also illustrated
in Figure 3. The seasonal variation in α(t) is very similar to the seasonal variation in ν(t),
though amplitudes are different. However, it is much more clear that α(t) has two local maxima
and two local minima. 16 Again, the global maximum is achieved in July about two months
prior to the beginning of the US harvesting while the global minimum is achieved and the end
of the US harvesting. While the US soybean production accounts for 50-60% of world soybean
production other key producers are Argentina and, especially, Brazil which are both located
in the Southern Hemisphere. These countries account together for 20-30% of world soybean
production. The Brazilian soybean harvesting takes place in March and April and the local
maximum in Figure 3 is reached about two months earlier, while the local minimum is reached
during the ending of the Brazilian soybean harvesting. The seasonal pattern in convenience
yields is thus related to the supply of soybeans. When supplies are low convenience yields are
high, and vice versa. This is consistent with the theory of storage which predicts a negative
relationship between inventories and convenience yields. The basic idea is that the holding of
physical inventory of soybeans gives rise to a convenience yield from being able to profit from
temporary soybean supply shortages or keep a production process running especially when
inventories are low. Furthermore, the parameter estimates suggest that the soybean spot price
and the convenience yield are positively correlated with a correlation coefficient of 0.3979.
Following the above kind of reasoning, this is also consistent with the observation that when
supplies and inventories are scarce the equilibrium soybean spot price and the convenience
yield are simultaneously high; and vice versa when supplies and inventories are plenty.
On the other hand, supporters of the theory of storage often view the convenience yield as
an option to profit from temporary shortages of the particular commodity. This would suggest
a positive correlation between the convenience yield and the volatility; however, this view is
not supported by the parameter estimate of ρ 23 which is not significantly different from zero
and even estimated negative.
16 The global maximum of the seasonal function α(t) is reached on July 15 where the function value is 0.746,
while the global minimum is reached on October 30 where the function value is -0.723. A local maximum is
reached on February 7, and a local minimum is reached on April 16.
20

Inverse leverage effects
The correlation coefficient between the soybean spot price and the soybean volatility is estimated
to be 0.4078. This reflects that the volatility tend to be high when soybean spot prices
are high and, in which case, supplies/inventories tend to be scarce. While this seems only
natural and an intuitively appealing feature for commodities like soybeans, this is different
from equity markets where spot prices and volatilities are usually estimated to be negatively
correlated; in equity markets this phenomenon is often referred to as the leverage effect. This
is consistent with the observation that the volatility “smirk” in soybean option prices has the
opposite slope of that implied from equity options after October 1987, as described in our data
description. Note also that the estimated positive correlation between the spot price and the
volatility seems intuitively to support the observation that OTM call option should trade at
higher implied volatilities. In particular, in order to have a positive payoff and value effect on
the call option, the underlying spot price must increase and, since the volatility in this case
also tends to increase, this will magnify the upward potential of the particular option (while
keeping downside risk limited).
Risk premia and investment returns
As seen from Table 3, in all cases the risk premia parameters are estimated with large standard
errors. Only the risk premium on volatility risk is significantly different from zero, and the
implications with respect to the long-run level for the seasonal adjusted volatility under the
risk-neutral measure, as reflected in market prices, and the true probability measure was
discussed earlier on.
While no strong inference can be drawn from the other risk premia
estimates, it seems relevant to briefly consider the implications of the parameter point estimates
for the implied excess returns on e.g. soybean futures positions. We will consider an investor
who exposes himself to soybean price risk by going long in futures contracts and at the same
time places an amount equal to the futures price on a safe bank account.
Since the drift
rate of the futures price is zero under the risk-neutral probability measure, the excess return
on this investment policy is described by the drift rate of the futures price under the true
probability measure. Using Ito’s lemma on the futures price expression in equation (10), and
by comparison with the risk-neutral dynamics of the futures price in (14), it is thus seen that
the expected excess return on this investment position is
[ ] dFt (T )
(
)
E t = λ P e ν(t) + λ δ σ δ e ν(t) D(t; T ) + λ v σ v B(t; T ) v t (29)
F t (T )
21

where D(t; T ) and B(t, T ) are described in (11) and (12). Assume that we are looking at the
futures position at May 17 in a given year; this date is chosen as a point in time t where
the seasonal component in volatility is zero (i.e. ν(t) = 0). Also, assume that the spot price
volatility is √ v t = 0.20 (i.e. v t = 0.04). If the involved futures contract is very short (i.e. t = T ),
we have that D(t; t) = B(t; t) = 0 and that the position is only exposed to spot commodity
price risk. Inserting the parameter point estimates in Table 3 into (29), the expected excess
return in this case is negative and equal to -2.51%. Longer term futures prices are negatively
related to the convenience yield (see equations (10) and (11)). Hence, since the risk premium
on convenience yield risk is estimated negative, longer term futures contracts will have a higher,
and possibly positive, overall risk premium when using the point estimates in Table 3. Also,
as seen by numerical inspection, futures prices at the point estimates are negatively related to
volatility. Since the risk premium on volatility risk is estimated negative, this also implies that
longer term futures contracts will have a higher, and possibly positive, overall risk premium.
However, when solving (11) and (12) for D(t; T ) and B(t, T ), and by insertion in (29), we find
that the overall expected excess return is -1.95% if the involved futures contract has time to
maturity equal to half a year. Likewise, the overall expected excess return is -1.77% if the
involved futures contract has time to maturity equal to one year. 17 In all the cases considered
above, the risk premium on the long futures investment position is negative. Furthermore, since
the futures price is expected to drift downwards this is also a case where the futures price is
above the expected future spot price, and thus a situation usually referred to as “contango.” 18
On the other hand, the only risk premia parameter which is significantly different from zero
is λ v and if we set λ P = λ δ = 0, normal backwardation would be the situation. It is thus
concluded that no strong inference can be drawn from the risk premia point estimates with
respect to the quantitative implications as well as the qualitative implications for expected
soybean investment returns.
Structure of measurement error covariance matrix
The measurement noise term added in the state space specification of the model is estimated
to have a covariance matrix as implied by the stated correlation matrix of measurement errors
in Table 3. Thus, using the one-to-one relationship between the tabulated correlation matrix
17 By solving (11) and (12) numerically, we in the particular cases find that B(t; t + 0.50) = -0.03151, D(t; t +
0.50) = −0.41071, B(t; t + 1.00) = -0.03475, and D(t; t + 1.00) = -0.68403.
18 see e.g. Hull (2000, p. 74) for this definition of contango versus normal backwardation.
22

(with the relevant estimated standard deviation in the diagonal), it is straightforward to
obtain the relevant covariance matrix, and vice versa. Moreover, applicable standard errors
on the covariance matrix can be obtained by using the result stated in section 2.4 that N ˆΓ
is Wishart distributed. In particular, having N=753 observation dates the standard errors on
the standard deviation estimates in the diagonal can be obtained as 0.0364 (= 1/ √ 753) times
the true standard deviation parameter (which, as usual, may by approximated by the point
estimate). Since we are only interested in the overall structure of the measurement errors, we
have only tabulated the point estimates on the covariance matrix in Table 3, and, in addition,
we have chosen to represent the covariance matrix in the form of the implied correlation matrix
in order to focus on a normalized measure of how the model errors co-vary which potentially
could give some indications and directions for how to improve the formal underlying futures
and option pricing model.
The measurement noise error vector has dimension ten, as described in the data description.
The first four entries in the vector are related to option prices in the following order: a short
maturity in-the-money call option, a short maturity out-of-the-money call option, a medium
maturity at-the-money call option, and a long maturity at-the-money call option. In general,
the longer the maturity the less precise is the formal model in providing at-the-mark option
prices, as can be inferred from the higher standard deviations on the measurement error terms
for long maturity contracts. Also, it is evident that the measurement errors on the medium
maturity and long maturity call options are positively correlated so that if the medium maturity
call option is underpriced by the theoretical option pricing model then the long maturity call
option contract will also tend to be underpriced; the point estimate of the relevant correlation
coefficient is 0.7310. It is likewise seen that the measurement errors on the short in-the-money
call option and out-of-the-money call option are negatively correlated. This could be consistent
with, and thus indicative of, an even steeper empirical volatility surface than produced by the
formal option pricing model with parameters as in Table 3 since a relatively flat model volatility
curve would tend to produce undervalued in-the-money options whenever out-of-the money
options are overvalued, and vice versa.
The last six entries in the measurement noise errors correspond to relative errors on futures
prices with different times to maturity. Number five entry relates to the shortest time to
maturity futures contract observed with measurement error, number six relates to the second
shortest time to maturity futures contract observed with measurement error, and so on. In
the estimation approach we have assumed that futures prices on a short maturity contract
23

and the longest available futures contract are observed without measurement error. Hence, it
is not surprising that short and long contracts are seemingly fitted better by the model than
medium maturity contracts, as reflected in the higher standard deviations on the imposed
measurement errors for medium maturity contracts. Note that the standard deviations on the
individual error terms on futures prices are estimated of the same numerical magnitude as in
Schwartz (1997), Schwartz and Smith (2000), and Sørensen (2001) who use only commodity
futures data and Kalman filtering estimation approaches. For example, Sørensen (2001) provides
results for soybean futures where the measurement errors are assumed uncorrelated and
with identical standard deviation which he estimates to be 0.187. As apparent from the correlation
matrix in Table 3, especially measurement errors for contracts that have maturities close
to each other are highly positively correlated while the correlations decrease systematically for
contracts having increasingly dispersed maturities.
Finally, the estimated correlations indicate slight co-variation between option prices and
futures prices measurement errors. In particular, the measurement errors related to short maturity
options have higher correlations with the measurement errors related to short maturity
futures while the medium/long maturity option measurement errors are more correlated with
measurement errors on futures prices of similar maturities. The correlations are in general
positive, however, no clear pattern is evident since the measurement error related to short maturity
in-the-money call options is negatively correlated with the measurement error on futures
prices of similar maturity while the short maturity out-of-the-money call option measurement
error at the same time is positively correlated with the same futures price measurement error.
Structural changes
In order to provide insights into whether the basic dynamic commodity price and contingent
claims pricing model is robust over time, we have estimated the model for two equally
long non-overlapping sub-periods. The relevant sub-samples thus cover the periods: October
1984 to November 1991 and November 1991 to March 1999. Rather than purely quantitative
econometric considerations with respect to diagnostic checking of the estimated model, this
investigation is especially motivated by qualitative considerations such as: Does seasonal patterns
change systematical over time? And, are correlation structures qualitatively robust over
time?
Seasonal patterns may vary over time due to especially production and storage innovations
24

as well as changes in government policies with respect to intervention in grain markets. For
example, Frechette (1997) analyzes the “flattering” of the soybean futures profile up through
the 1970s where the Brazilian soybean production increased rapidly and changed the seasonal
pattern in world soybean production. However, such dramatic changes in world soybean
production patterns have not occurred in the time periods relevant for our investigation. Also,
although US government policies prevailing in the 1970s and 1980s provided a floor under
grain prices, soybeans had a relatively low support level and, thus, also no dramatic changes in
government intervention policies with respect to soybeans seem to have occurred in the relevant
estimation periods. In sum, it is not obvious on beforehand what kind of parameter instability
one would expect to find due to changes in soybean markets conditions when splitting the data
into sub-samples.
The sub-period estimation results are tabulated in Table 4 and Table 5, respectively.
[ INSERT TABLE 4 AND TABLE 5 ABOUT HERE ]
We will be brief in our discussion of the differences between the estimates in the two subperiods
and will mainly point out and focus on differences in the persistence of shocks, seasonal
patterns, and correlations of relevance to the above full-sample discussion of the relation to
the theory of storage and the “inverse leverage effect.”
It may first be noted that volatility shocks and convenience yield shocks seemingly are more
persistent in the first sub-period than the second sub-period, as reflected in the estimates of
the mean-reversion parameters κ and β. Thus, the estimates of κ indicate that, under the
risk-neutral measure, the half-lives of shocks to volatility equal 0.940 years in the first subperiod
and 0.160 years in the second sub-period (while the full-sample analog is 0.305 years,
as discussed earlier). While this is true under the risk-neutral measure, as reflected in options
and futures prices, the relevant mean-reversion parameter under the true probability measure
is given by κ − λ v σ v . Hence, under the true probability measure the persistence of shocks to
the volatility is much more similar across the two sub-periods, and relevant half-lives are 0.141
years and 0.093 years, respectively, as opposed to 0.137 years for the full-sample estimation
results.
In order to see more clearly the estimated seasonality patterns in volatilities across the two
sub-periods, we have graphed the relevant seasonal components in Figure 4.
[ INSERT FIGURE 4 ABOUT HERE ]
25

The figure shows the estimated seasonal function ν(t) for the full-sample as well as the estimated
seasonal function for the first sub-period, ν 1 (t), and the estimated seasonal function
for the second sub-period, ν 2 (t). The estimated seasonal functions seem very similar with
amplitudes of similar magnitude and with minima and maxima attained at about the same
time of the year. The amplitude is slightly higher in the second sub-period and, if not simply
due to randomness, this could be due to the above-mentioned price floors maintained by the
US government programs up through the 1980s.
I Figure 5 we have similarly illustrated the seasonal patterns of the mean-reverting level of
the convenience yield as estimated in the first sub-period, α 1 (t), the second sub-period, α 2 (t),
as well as for the full sample, α(t).
[ INSERT FIGURE 5 ABOUT HERE ]
While the estimated functions attain minima and maxima at the same times of the year,
the amplitudes seem very different. However, though the amplitude for the first sub-period
estimation is much larger, it may be noted that the mean-reversion tendency is at the same
time much lower than for the second sub-period, as reflected in the β estimates. Hence, the
overall seasonal effect on the level of the convenience yield may not differ very substantially
across the two sub-periods.
Finally, it is seen from Table 4 and Table 5 that the estimated correlation coefficients (ρ 12 ,
ρ 13 , and ρ 23 ) are very similar across the two sub-periods. Thus, the conclusion that, e.g.,
convenience yields do not behave like timing options is robust over time since in both subperiods
the correlation coefficient, ρ 23 , between the convenience yield and volatility is close to
zero and, in fact, consistently estimated slightly negative. Also, the “inverse leverage effect” is
robust over time since the correlation coefficient, ρ 13 , between the spot price and the volatility
is consistently estimated positive and of the same magnitude across the sub-periods.
4 Conclusions
In the paper we have set up a stochastic volatility model of commodity price behavior and
estimated the model parameters using panel-data of both soybean futures and soybean options.
The estimation approach thus use the information in both time series characteristics of the
evolvement of soybean product prices as well as the information inherent in futures price
structures and option price structures, such as implied volatility surfaces.
26

Besides estimating model parameters, which among other things facilitates implementation
of hedging strategies and related applications of the specific model, we find a clear seasonal
pattern in both volatilities and convenience yields. Also, our results are to some extent consistent
with the theory of storage in the sense that the estimations suggest that convenience yields
tend to be low when inventories/supply is high, and vice versa. However, our results do not
support the view that convenience yields are like timing options since convenience yields are
not positively correlated with volatility. Furthermore, we find a positive correlation between
spot prices and volatility, as opposed to the so-called leverage effect in equity markets, and
this is consistent with a different sloping “smirk” in the implied volatility curve for soybean
options.
27

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32

Appendix
Lemma 1 Let p t = log P t . The conditional means and conditional variances of the process
X t = (p t , δ t , v t ), described by the stochastic differential equation system (1), (2), and (3), are
given by
where
(
E t [X t+∆ ] = Φ(t + ∆) X t +
∫ t+∆
t
)
Φ −1 (u)l(u) du
(∫ t+∆
)
Var t [X t+∆ ] = Φ(t + ∆) E t [v u ] Φ −1 (u)Σ(u)Φ −1 (u) ′ du Φ(t + ∆) ′ (31)
t
⎛ ⎞ ⎛
⎞
r
e ν(u) ρ 12 σ δ e ν(u) ρ 13 σ v e 1 2 ν(u)
l(u) = ⎜ α(u) ⎟
⎝ ⎠ , Σ(u) = ⎜ ρ
⎝ 12 σ δ e ν(u) σδ 2eν(u) ρ 23 σ v σ δ e 1 2 ν(u)
⎟
⎠
θ
ρ 13 σ v e 1 2 ν(u) ρ 23 σ v σ δ e 1 2 ν(u) σv
2
and where Φ is the unique solution to the matrix differential equation
with
dΦ(u)
du
(30)
= L(u)Φ(u) , Φ(t) = I (32)
⎛
⎞
0 −1 − 1 2 + λ P e 1 2 ν(u)
L(u) = ⎜ 0 −β λ
⎝
δ σ δ e 1 2 ν(u)
⎟
⎠ .
0 0 −κ + λ v σ v
In particular, the conditional expectations entering the right hand side integral of (31) are
described by (30) and given by E t [v u ] = v t e −κ(u−t) + θ κ (1 − e−κ(u−t) ).
Proof:
The dynamics of X t = (p t , δ t , v t ) can be written in matrix form as
dX t = (L(t)X t + l(t)) dt + √ v t σ(t) dW t (33)
where σ(t) = diag[e 1 2 ν(t) , σ δ e 1 2 ν(t) , σ v ] is a diagonal matrix with the three arguments in the
diagonal and zeros off the diagonal. By using Ito’s lemma, and the definition of Φ in (32), it
is seen that X can be characterized as the solution to (s ≥ t)
( ∫ s
∫ s
)
X s = Φ(s) X t + Φ −1 √
(u)l(u) du + vu Φ −1 (u)σ(u) dW u . (34)
Note that ∫ s
t
t
t
√
vu Φ −1 (u)σ(u) dW u , s ≥ t, is a martingale starting at zero at time t. Hence,
by taking conditional expectations in (34) (and setting s = t + ∆), one obtains (30). Furthermore,
since the two first terms on the right hand side of (34) are F t -measurable, the result
in (31) follows by an application of e.g. Karatzas and Shreve (1991), Proposition 2.17, p. 144.
33

Table 1: Summary statistics for soybean futures, 1984-1999.
Contracts
Days to maturity
Settlement prices
Number of
observations Mean Mean Std. deviation
All 6024 209.53 627.49 86.08
Grouped into time to maturity:
1. Maturity 753 28.60 622.86 97.13
2. Maturity 753 81.75 624.82 95.24
3. Maturity 753 132.52 626.84 92.24
4. Maturity 753 183.57 627.88 87.92
5. Maturity 753 234.00 628.75 83.81
6. Maturity 753 284.36 628.99 80.09
7. Maturity 753 337.77 629.18 76.38
8. Maturity 753 393.63 630.59 72.68
Grouped into expiration months:
January 893 215.46 622.51 81.96
March 877 210.84 628.02 85.24
May 872 212.17 634.03 87.13
July 883 214.26 638.37 91.18
August 812 197.62 632.63 90.16
September 808 197.84 621.02 84.49
November 879 216.55 615.77 80.09
34

Table 2: Summary statistics for soybean call options, 1984-1999.
Contracts
Days to maturity
Implied volatilities
Number of
observations Mean Mean Std. deviation
All 3765 71.95 19.99 6.60
Grouped into time to maturity and moneyness:
1. Maturity – ITM ∗ 753 40.50 18.88 6.53
1. Maturity – ATM ∗ 753 40.50 19.61 6.99
1. Maturity – OTM ∗ 753 40.50 21.58 7.19
2. Maturity – ATM ∗ 753 93.71 19.81 6.38
3. Maturity – ATM ∗ 753 144.69 20.06 5.51
Grouped into expiration months:
January 548 66.65 18.13 4.72
March 636 77.32 17.13 3.72
May 645 78.49 16.71 4.17
July 631 81.50 19.40 4.84
August 431 71.85 25.17 8.40
September 374 59.04 26.28 8.76
November 500 60.17 21.47 5.74
∗ The average call prices are: 30.58 for 1. Maturity – ITM, 15.68 for 1. Maturity – ATM, 8.31
for 1. Maturity – OTM, 25.25 for 2. Maturity – ATM, and 31.37 for 3. Maturity – ITM.
35

Table 3: Parameter estimates for soybean contracts, Full sample: Oct. 1984 – Mar. 1999.
Volatility κ: 2.2708 θ : 0.1542 σ v : 0.7414 ν 1 : −0.4096 ν ∗ 1 : −0.2626 ν 2 : 0.2250 ν ∗ 2 : 0.1293
parameters: (0.2654) (0.0085) (0.0299) (0.0387) (0.0483) (0.0229) (0.0223)
Convenience yield β : 0.8145 σ δ : 0.9933 α 0 : 0.0612 α 1 : −0.2683 α ∗ 1 : 0.1915 α 2 : 0.2338 α ∗ 2 : 0.2689
parameters: (0.2100) (0.1345) (0.0086) (0.0276) (0.0130) (0.0255) (0.0600)
Correlations and risk ρ 12 : 0.3979 ρ 13 : 0.4078 ρ 23 : −0.0784 λ P : −0.6274 λ δ : −0.1282 λ v : −3.7829
premia parameters: (0.0110) (0.0461) (0.0772) (1.3001) (1.2286) (1.5610)
36
Correlation matrix of measurement errors (standard deviation estimates in diagonal):
⎛
⎜
⎝
⎞
0.9606
−0.3075 0.8040
0.0489 0.2150 2.4068
0.0164 0.1373 0.7310 3.4679
−0.2480 0.1719 0.1118 0.1053 0.0180
−0.1871 0.1153 0.1435 0.1549 0.8947 0.0265
−0.1236 0.0850 0.2187 0.2398 0.7175 0.9137 0.0279
−0.0638 0.0566 0.2922 0.3084 0.5302 0.7388 0.9238 0.0254
⎟
−0.0688 0.0755 0.2306 0.2765 0.3838 0.5751 0.7466 0.8870 0.0193 ⎠
−0.1199 0.0808 0.1573 0.2042 0.2053 0.3663 0.4892 0.6014 0.8191 0.0116
Log-likelihood function value: 8468.59

Table 4: Parameter estimates for soybean contracts, 1. half: Oct. 1984 – Nov. 1991.
Volatility κ: 0.7373 θ : 0.1103 σ v : 0.7626 ν 1 : −0.3825 ν ∗ 1 : −0.1448 ν 2 : 0.2461 ν ∗ 2 : 0.0861
parameters: (0.5038) (0.0110) (0.0282) (0.0371) (0.0423) (0.0332) (0.0186)
Convenience yield β : 0.3186 σ δ : 0.7385 α 0 : 0.0872 α 1 : −0.2103 α ∗ 1 : 0.1571 α 2 : 0.2300 α ∗ 2 : 0.2454
parameters: (0.1921) (0.0887) (0.0090) (0.0336) (0.0139) (0.0335) (0.0516)
Correlations and risk ρ 12 : 0.3549 ρ 13 : 0.4989 ρ 23 : −0.0638 λ P : −3.0441 λ δ : −1.9042 λ v : −5.4792
premia parameters: (0.0150) (0.1052) (0.0836) (2.0377) (1.6564) (1.8105)
37
Correlation matrix of measurement errors (standard deviation estimates in diagonal):
⎛
⎜
⎝
⎞
1.0093
−0.1240 0.8056
0.1506 0.2121 2.6630
0.0842 0.1804 0.6474 3.6959
−0.2235 0.1148 0.1393 0.0651 0.0142
−0.1720 0.0662 0.0958 0.0409 0.9347 0.0219
−0.1396 0.0662 0.1087 0.0256 0.8492 0.9599 0.0251
−0.1216 0.0656 0.1563 0.0505 0.7250 0.8476 0.9381 0.0244
⎟
−0.1333 0.0901 0.0660 0.0475 0.5411 0.6404 0.7345 0.8644 0.0206 ⎠
−0.1787 0.0821 −0.0131 0.0266 0.3112 0.3578 0.4195 0.5321 0.8027 0.0129
Log-likelihood function value: 4205.60

Table 5: Parameter estimates for soybean contracts, 2. half: Nov. 1991 – Mar. 1999.
Volatility κ: 4.3447 θ : 0.2572 σ v : 0.8736 ν 1 : −0.4185 ν ∗ 1 : −0.2587 ν 2 : 0.2193 ν ∗ 2 : 0.1421
parameters: (0.3286) (0.0144) (0.0351) (0.0190) (0.0268) (0.0108) (0.0103)
Convenience yield β : 1.1901 σ δ : 1.1621 α 0 : 0.0328 α 1 : −0.1875 α ∗ 1 : 0.1777 α 2 : 0.1489 α ∗ 2 : 0.1879
parameters: (0.0794) (0.1401) (0.0047) (0.0068) (0.0073) (0.0039) (0.0035)
Correlations and risk ρ 12 : 0.3972 ρ 13 : 0.5085 ρ 23 : −0.0409 λ P : −0.7322 λ δ : −1.8687 λ v : −3.5911
premia parameters: (0.0115) (0.0361) (0.1197) (2.1073) (1.9719) (2.8661)
38
Correlation matrix of measurement errors (standard deviation estimates in diagonal):
⎛
⎜
⎝
⎞
0.9013
−0.4926 0.7990
−0.1575 0.3329 1.9643
−0.1543 0.2437 0.7839 2.5335
−0.2784 0.1928 0.0113 0.0066 0.0213
−0.2068 0.1145 0.0766 0.0743 0.8746 0.0303
−0.1114 0.0506 0.1487 0.1832 0.6274 0.8831 0.0297
−0.0207 0.0071 0.2015 0.2410 0.3607 0.6371 0.8998 0.0254
⎟
−0.0263 0.0403 0.1731 0.1870 0.1877 0.4560 0.6987 0.8859 0.0178 ⎠
−0.0891 0.0948 0.1543 0.1252 0.0098 0.2526 0.4272 0.5733 0.8113 0.0107
Log-likelihood function value: 4642.37

1100.00
Futures prices in cents per bushel
1000.00
900.00
800.00
700.00
600.00
500.00
400.00
Jan-84 Jan-88 Jan-92 Jan-96 Jan-2000
Short maturity
Long maturity
Figure 1: Time series of soybean futures prices. The figure displays the futures prices on
the contracts with the shortest (1. Maturity) time to maturity and the longest (8. Maturity)
time to maturity over the full sample period from October 1984 to March 1999.
39

0.80
0.70
0.60
Implied volatilities
0.50
0.40
0.30
0.20
0.10
0.00
Jan-84 Jan-88 Jan-92 Jan-96 Jan-2000
Figure 2: Time series of implied volatilities on soybean call options. The figure
displays the implied volatilities on the at-the-money call option contracts with the shortest (1.
Maturity) time to maturity over the full sample period from October 1984 to March 1999.
40

1.00
0.75
0.50
0.25
0.00
- 0.25
α(t)
ν(t)
frag replacements
- 0.50
- 0.75
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Figure 3: The seasonal functions α(t) and ν(t). The figure displays the target seasonal
level of convenience yields, α(t), and the seasonal component in the soybean price volatility,
ν(t).
41

0.50
0.25
0.00
ν 1 (t)
ν 2 (t)
ν(t)
- 0.25
frag replacements
- 0.50
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Figure 4: The three volatility seasonal functions ν(t), ν 1 (t), and ν 2 (t). The figure
displays the seasonal component in the soybean price volatility for the full sample, ν(t), the
first sub-period, ν 1 (t), and the second sub-period, ν 2 (t)
42

2.00
1.50
1.00
0.50
0.00
- 0.50
α 1 (t)
α(t)
α 2 (t)
frag replacements
- 1.00
- 1.50
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Figure 5: The three convenience yield seasonal functions α(t), α 1 (t), and α 2 (t). The
figure displays the relative seasonal level of convenience yields for the full sample, α(t), the
first sub-period, α 1 (t), and the second sub-period, α 2 (t)
43